Integrand size = 18, antiderivative size = 63 \[ \int x^m \left (c x^2\right )^p (a+b x)^n \, dx=\frac {x^{1+m} \left (c x^2\right )^p (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,1+m+2 p,2+m+2 p,-\frac {b x}{a}\right )}{1+m+2 p} \]
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Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {15, 68, 66} \[ \int x^m \left (c x^2\right )^p (a+b x)^n \, dx=\frac {x^{m+1} \left (c x^2\right )^p (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,m+2 p+1,m+2 p+2,-\frac {b x}{a}\right )}{m+2 p+1} \]
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Rule 15
Rule 66
Rule 68
Rubi steps \begin{align*} \text {integral}& = \left (x^{-2 p} \left (c x^2\right )^p\right ) \int x^{m+2 p} (a+b x)^n \, dx \\ & = \left (x^{-2 p} \left (c x^2\right )^p (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n}\right ) \int x^{m+2 p} \left (1+\frac {b x}{a}\right )^n \, dx \\ & = \frac {x^{1+m} \left (c x^2\right )^p (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (-n,1+m+2 p;2+m+2 p;-\frac {b x}{a}\right )}{1+m+2 p} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int x^m \left (c x^2\right )^p (a+b x)^n \, dx=\frac {x^{1+m} \left (c x^2\right )^p (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,1+m+2 p,2+m+2 p,-\frac {b x}{a}\right )}{1+m+2 p} \]
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\[\int x^{m} \left (c \,x^{2}\right )^{p} \left (b x +a \right )^{n}d x\]
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\[ \int x^m \left (c x^2\right )^p (a+b x)^n \, dx=\int { \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{n} x^{m} \,d x } \]
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\[ \int x^m \left (c x^2\right )^p (a+b x)^n \, dx=\int x^{m} \left (c x^{2}\right )^{p} \left (a + b x\right )^{n}\, dx \]
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\[ \int x^m \left (c x^2\right )^p (a+b x)^n \, dx=\int { \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{n} x^{m} \,d x } \]
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\[ \int x^m \left (c x^2\right )^p (a+b x)^n \, dx=\int { \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{n} x^{m} \,d x } \]
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Timed out. \[ \int x^m \left (c x^2\right )^p (a+b x)^n \, dx=\int x^m\,{\left (c\,x^2\right )}^p\,{\left (a+b\,x\right )}^n \,d x \]
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